In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f ( z ) = a z + b c z + d {\displaystyle f(z)={\frac {az+b}{cz+d}}}
of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C). Together with its subgroups, it has numerous applications in mathematics and physics. Möbius geometries and their transformations generalize this case to any number of dimensions over other fields. Möbius transformations are named in honor of August Ferdinand Möbius; they are an example of homographies, linear fractional transformations, bilinear transformations, and spin transformations (in relativity theory).
Overview Möbius transformations are defined on the extended complex plane C ^ = C ∪ { ∞ } {\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}} (i.e., the complex plane augmented by the point at infinity). Stereographic projection identifies C ^ {\displaystyle {\widehat {\mathbb {C} }}} with a sphere, which is then called the Riemann sphere; alternatively, C ^ {\displaystyle {\widehat {\mathbb {C} }}} can be thought of as the complex projective line C P 1 {\displaystyle \mathbb {C} \mathbb {P} ^{1}} . The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of C P 1 {\displaystyle \mathbb {C} \mathbb {P} ^{1}} as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, and is sometimes denoted Aut ( C ^ ) {\displaystyle \operatorname {Aut} ({\widehat {\mathbb {C} }})} . The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds. In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory. Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations. Möbius transformations can be more generally defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions.
Definition The general form of a Möbius transformation is given by
f ( z ) = a z + b c z + d , {\displaystyle f(z)={\frac {az+b}{cz+d}},}
where a, b, c, d are any complex numbers that satisfy ad − bc ≠ 0. In case c ≠ 0, this definition is extended to the whole Riemann sphere by defining
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